The Geometrical Works of Abu Sa'id Al-Dartr Al-Jurjant
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SCIAMVS 2 (2001), 47-74 The Geometrical Works of Abu Sa'Id al-DarTr. al-JurjanT Jan P. Hogendijk Department of Mathematics University of Utrecht For Abu Max I Abii Sacid al-Qarir al-JurjanI Almost nothing is known about the life of the geometer Abu Sa<-rdal-I;)arir ("the blind") al-JurjanI. His name indicates that he originated from the town of Jurjan, that is present-day Gargan near the Caspian Sea in Iran. The most complete form of his name appears in one version of al-BirunI's Extraction of Chords [20, p.13]1 as Abu Sa<-rdMul).ammad ibn AlI al-I;)arir al-JurjanI. In [19, p. 27], Suter confused the geometer, Abu Sa<"fdal-I;)arir al-JurjanI, with the philologist, Abu Sa<-rdal-I;)arir Al).mad ibn Khalid, who died in 895 CE. 2 Consequently, al-JurjanI is listed as a ninth-century mathematician by Sezgin [17, vol. 5, pp. 263-264, vol. 6, p. 159] and by Matvievskaya and Rozenfeld [13, vol. 2, p. 76, no. 48]. In the Geometrical Problems, published below, al-JurjanI mentions Abu Abdallah al-ShannI, 3 who lived in the second half of the 10th century CE. This reference confirms that the geometer Mul).ammad ibn AlI al-JurjanI is not identical to the philologist Al).mad ibn Khalid. We will see below that al-JurjanI must have flourished around 1000 CE. 4 Two treatises by al-JurjanI have come down to us, both of considerable historical interest. 1. His Geometrical Problems begin with a construction of a neusis by means of a parabola and hyperbola. This construction is a variation of a construction in Book IV of the Mathematical Collection of Pappus of Alexandria. Then follow solutions of two problems related to the trisection of the angle, and a theorem about the altitude of a triangle. The Geometrical Problems is addressed to an anonymous mathematician, who will be identified below as al-BirunI. One of 1 0n al-BfriinI (973-1048) see for example [17, vol.6, pp.261-276]. 2 0n Abu Sa'Id al-Qarir Agmad ibn Khalid see [17, vol.8, pp.167-168]. 3 0n al-ShannI see [17, vol. 5, p. 352]. Around 970 CE he quarreled with Abii'l-Jiid in connection with the regular heptagon, see [7, pp. 243-244]. 4 In [5, p. 331] Hermelink stated that al-JurjanI lived in the second half of the 10th century CE. 48 Jan P. Hogendijk SCIAMVS 2 al-JurjanI's constructions is related to the "Problem of Alhazen" in the Optics of Ibn al-Haytham. Thus the Geometrical Problems of al-JurjanI are connected to some of the highlights of Arabic-Islamic geometry. 2. Most of al-JurjanI's Extraction of the Meridian Line is based on a lost work Analemma by Diodorus of Alexandria, a Greek expert on sundial theory who lived in the first century BCE. The only other extant trace of Diodorus' Analemma is Chapter 20 in al-BirunI's Exhaustive Treatise on Shadows. Al-JurjanI's treatise on the meridian line was published in 1922 in a free German translation by Schoy [18], but the Arabic text has not hitherto been edited. The Geometrical Problems seem to be completely unknown to the literature. The only other traces of geometrical work by al-JurjanI are two short proofs which are cited by al-BirunI in the Extraction of Chords.5 The purpose of this paper is to publish and translate the Geometrical Problems and the Extraction of the Meridian Line, in order to make the entire extant work of al-JurjanI available for further historical analysis. I begin with a brief description of the contents of the Geometrical Problems, and a brief summary of the Extraction of the Meridian Line. This is followed by a section on manuscripts and editorial procedures, and edited Arabic texts and English translations of the two treatises. II The Geometrical Problems of al-J urjanI Al-JurjanI begins his Geometrical Problems with a preliminary construction which I render in modernized notation (Figure 1). The text and translation can be found below. Given: three points A, B, G on a circle, and a segment HZ. Required: to construct a straight line AED which intersects segment BG at E and the circle at D in such a way that ED = H Z. 6 In ancient Greek geometry, this type of construction was called a neusis, that is the insertion of a straight segment (DE) of given length (HZ) between two given lines which may be straight or curved (in this case segment BG and arc BG), in such a way that a given point (A) is on the segment or on its rectilinear extension. Al-JurjanI bisects BG at T and he draws a perpendicular to BG through T. He draws AL parallel to BG to meet the perpendicular at L, he drops perpendicular 5 The Arabic text is available, on the basis of the Patna manuscript, in [1, no. 1, pp. 8, 23] and [2, pp. 40, 57-48], for a facsimile of the Patna manuscript of the first proof see [2, p. 30]. A German translation of the two proofs, on the basis of the Leiden manuscript, can be found in [20, pp. 13, 15]. 6 1n this paper I use a notation such as ED in the ancient Greek way. Thus ED may indicate a straight line segment with endpoints D and E, but also the length of this segment. SCIAMVS 2 The Geometrical Works of Abu Sa'Id al-Qarir al-JurjanI 49 s N Figure 1 AK onto BG, and he chooses point Son KA extended such that AS= AK. 2 He then defines point N on line LT such that LN · HZ = i:BG • He assumes 2 HZ· AK< :tBG , whence LN > LT. He then draws one branch 1{ of the equilateral (orthogonal) hyperbola through K with centre A and transverse axis SK, and the parabola P with vertex N, axis NL and parameter (latus rectum) HZ. Al-Jurjan1 supposes that P and 1{ intersect at point M, and he drops perpendic ulars ME onto BG and M X onto AK extended. Let M X intersect LN at O. Since Mis on 1{, we have by Apollonius, Conics 1:21,7 MX 2 = SX, KX. But MX = EK, henceEA 2 = EK 2 +AK 2 = SX·KX+AK 2 = (AX 2 -AK 2 )+AK 2 = AX 2 , so EA= AX. Since Mis on the parabola P with latus rectum HZ, we have by Conics 1:118 M0 2 =ON· HZ. But MO= ET, and by definition GT 2 = LN · HZ. Thus by 2 2 subtraction BE· GE= GT - ET =LO· HZ= AX· HZ= AE ·HZ.From the geometry of the circle, BE· GE= AE · ED. Hence ED= HZ as required. Al-Jurjanl's construction can be seen as a variation on a construction in the end of Book IV of the Mathematical Collection of Pappus of Alexandria (ca. 300 CE). 9 Pappus solves the same problem by means of two conic sections P' and H', which 7 Putting XM = y, KX = x, SK= c we obtain the modern equation y 2 = x(x+c) of the hyperbola. 8 Putting OM= Y1, NO= x1, HZ= p we obtain the modern equation yf = px 1 of the parabola. 9 Pappus divides the construction into three propositions, see [23, vol. 1, pp. 231-235], [15, vol. 1, 50 Jan P. Hogendijk SCIAMVS 2 are the images of 'P and 1t under the translation of the plane which maps A to K .10 The perpendicular projection of the intersection M' of 'P' and 1t' on line BG is of course point E in Figure 1. The similarity with Pappus suggests that al-JurjanI's construction is ultimately of ancient Greek origin. We now turn to the diorismos of the problem, that is to say, the necessary and sufficient condition for the existence of a solution. 11 Al-JurjanI wrongly states that 2 the diorismos is HZ · AK :S f BG . Actually, this condition is necessary but not 2 sufficient. If HZ· AK = f BG , the problem can only be solved if K is the midpoint of BG. Figure 2 is drawn for the case where K is not the midpoint of BG and 2 HZ· AK is slightly less than f BG , so that N is a little below T. In this case 1t and 'P do not intersect, and the problem has no solution. L >----........_A H Z p Figure 2 If K is not the midpoint of BG, the correct diorismos has the form HZ :S C for some segment C with C · AK < f BG 2 . If HZ = C, the curves 'P and 1t are tangent, and by analyzing this case one can in principle find a construction of C from AK, BK and BG. This problem is by no means easy: the construction of line AED itself is equivalent to the solution of an irreducible algebraic equation of degree 4, and the length of C is the root of an irreducible cubic equation. By means of a different construction, lbn al-Haytham and al-Mu'taman ibn Hud (died 1085) found C as a normal from a point to a certain hyperbola. 12 pp. 298-303, corrected in vol. 3, pp. 1231-33]. 10 1 use the concept of "translation" for sake of brevity. Of course I do not want to suggest that Pappus or al-Jurjan1 had the modern concept of a translation of the plane.